138 XIII INTEGRATION
Now the sets Apt9tf are closed (3.11.4), hence BM?r is integrable (13.8.7). More-
over, by hypothesis, for each integer r ^ 1 we have ^/f) Bw>r\ = 0, and the
sequence (B^Xg! is decreasing, so that by (13.8.7) we have lim yu(Bn>r) = 0.
n-»oo
Hence there exists an integer nr such that n(Bnrtf) g s/2r+2. Let B = (J BMr,r;
r
the set B is integrable, and we have //(B) g f] s/2r+2 = Je by (13.8.7). The
r=l
set C = K0 - B is also integrable, and by definition the sequence (fn \ C)
converges uniformly to f\ C. Hence we may take K' to be any compact
subset of C such that n(C - K') ^ & (13.7.9).
(13.9.11) Let (fn) be any sequence of measurable functions on X with values in
R. Then the functions inffn> sup/M, lim sup/M and lim inffn are measurable.
n n ./i-* oo /i-* oo
For sup/, is the limit of the increasing sequence of functions gn = sup ft,
n l£i£n
which are measurable (13.9.7); and lim sup/n is the limit of the decreasing
w* oo
sequence of functions sup/n+p, which are measurable, by what has just been
p*o
proved.
(13.9.11.1) The propositions (13.9.6), (13.9.7), (13.9.9), and (13.9.11) remain
valid when the word "measurable" is replaced throughout by "universally
measurable." On the other hand, a function cannot be universally measurable
unless it is everywhere defined, because any nonempty subset N of X has
measure //(N) ^ 0 if for example \JL is the Dirac measure at a point of N.
In particular (12.7.8), every function belonging to«/ (resp. ff1) is universally
measurable. If / is any lower semicontinuous function on X, then by
writing
/ = inf (sup (/, - n)} and sup (/, - n) = ~ n + sup (/ + n, 0)
n
and using (13.9.6), it follows that
(13L9.11.2) Every (upper or lower) semicontinuous function on X with values
in R is universally measurable.
The same will be true of all functions obtained from semi-continuous
functions by repeated applications of the operations in (13.9.6), (13.9.7), and
(13.9.11).ictions u \ Ln is continuous.